Fundamental theoretical calculations of molecular properties, such as their mode of bonding, are based on the following mathematical procedure: Atomic orbitals, i.e., certain spatial distributions of electrons in atoms (hereafter sometimes referred to as "AO's" or an "AO") must be combined to form molecular orbitals, i.e., certain spatial distributions of electrons in molecules (hereafter sometimes referred to as "MO's" or an "MO") according to rigorous and complex mathematical rules derived from the discipline of group theory and pertaining to certain symmetry properties of the AO's and the molecule in question. The present invention furnishes an apparatus that makes it possible to determine the correct AO combinations and to reject all possible incorrect ones visually by mechanical means.
In general, the atomic orbitals of the non-central atoms are of the so-called s and p types. With reference to the valence electrons in the outer electron shells of atoms, there is always one s orbital, and there may also be three p orbitals. As is known, the s type AO's are spherically symmetrical, and therefore are schematically represented as spherical bodies. They may be distinguished as having either a positive or negative wave function, which can be represented by appropriate indicia on the s type AO models.
The p type AO's can be represented as symmetrical bodies having two lobes aligned on a common axis and joined at their inner ends. The lobe portions are most commonly shown as egg-shaped, but they can also be represented as spherical or ellipsoidal. The surface boundaries of the lobe portions of the p type AO's are mathematically defined, but can be represented without such exactitude for model purposes. Each lobe of the p type AO has a different wave sign (positive or negative), and therefore appropriate indicia can be applied to the opposed lobes to indicate the positive or negative nature of the wave functions. The p type AO's are usually arranged in several orientations, with reference to Cartesian axes (x, y, and z), which are also mathematically defined. Commonly used nomenclature is p.sub.x, p.sub.y, and p.sub.z atomic orbitals.
In the series of atomic orbitals, the next group is referred to as the d AO's. There are five d type AO's, in relation to electron charge distribution and orientation. Four have the same geometric shape, differing only in orientation, and are conventionally represented as symmetrical bodies having four lobes paired on common axes and with their inner end portions joined at a common center. The lobes, as indicated with respect to the p orbitals, may be represented as egg-shaped, ellipsoidal, or spherical. They are more commonly shown as egg-shaped lobes, but the exact shape for model purposes is not important. The adjacent lobes of the d orbitals of the four-lobe type, are of alternate wave sign (positive or negative), the axial pairs being of the same sign. Therefore, appropriate indicia may be applied to such models. With reference to the Cartesian axes (x, y, z) the four-lobed d AO's are commonly designated as d.sub.xy, d.sub.xz, d.sub.yz, and d.sub.x 2.sub..sub.-y 2. The fifth atomic orbital of the d type has a different shape, being referred to as the d.sub.z 2. It is represented as a two-lobe body, similar to that of the p AO's, the lobes being on a common axis and having their inner ends joined. In addition, however, a torroid is represented as surrounding the inner portions of the lobes. Such representation may be conventionalized, viz. lobe portions connected by a ring portion. The lobe portions have the same wave sign (positive or negative), while the wave sign of the toroid or ring portion is the opposite of the lobe portions. Therefore appropriate indicia indicating the differing wave signs can be applied.
Such model representations, of s, p, and d type AO's are wellknown to chemists, being described in standard treatises. See, for example, the following:
H. h. sisler, "Electronic Structure, Properties, and the Periodic Law", Reinhold Publishing Corp., New York, 1963, Ch. 1, especially pp. 26-27; PA1 F. a. cotton and G. Wilkinson, "Advanced Inorganic Chemistry", Second Ed., Interscience Publishers, New York, 1966, Ch. 1, especially p. 17; and PA1 E. grunwold, "Chemical Bonding and Structure," Ratheon Education Co., 1968, Ch. 2, especially pp. 16-18. PA1 According to bonding theory, the AO's above the d type, are of negligible importance with respect to peripheral atoms in a molecule, and need only be considered for a central atom when they are present among its valence orbitals. However, the shape of such AO's identified as the f, g, and h type AO's, are also mathematically determined, and are capable or representation as symmetrical bodies. See, for example: PA1 H. e. white, "Introduction to Atomic Spectra", McGraw-Hill Book Co., Inc., New YOrk, k1934, Ch. IV, especially p. 63; and PA1 M. karplus and R. N. Porter, "Atoms and Molecules", W. A. Benjamin, Inc., 1970, Ch. 3, especially pp. 132-135.
Prior to the present invention, a knowledge of molecular bonding concepts, group theory, quantum mechanics, and related subjects was required to determine the permitted electron distributions within molecules. It was not possible for a chemist or student of chemistry to generate the MO's appropriate for molecular bonding without a firm knowledge and working ability of advanced mathematical concepts and techniques. Further, it is difficult for the non-theoretically oriented person to visualize the various permitted and non-permitted atomic orbital arrangements in three dimensions. Moreover, such visualization is complicated by the fact that with molecules containing more than two atoms, it is possible to envision more atomic orbital arrangements than are permitted from bonding theory: thus, the terminology "permitted" and "non-permitted" AO arrangements. Consequently, there is a practical need for a simple 3-dimensional apparatus that will assist the average chemist or chemistry student in selecting visually by mechanical means the "permitted" arrangements of the AO's of non-central atoms, while at the same time excluding the non-permitted arrangements.
The system of this invention and the apparatus applying it provides the means for such a mechanical selection by introducing models of central atomic orbitals (AO's) as generators for constructing those arrangements of the non-central AO's which represent the permitted molecular orbitals (MO's) of the molecule, the central generators being used in the order of increasing AO complexity: s, p, d, etc. This concept and procedure is believed to be completely novel. In fact, as far as applicants are aware, no prior art apparatus at all is known for determining by mechanical means the permitted and non-permitted molecular orbitals.
The molecular apparatus system of this invention is advantageously provided in the form of a kit, which can be used to generate the permitted molecular orbitals of hundreds of molecules (although, of course, not for all molecules). In accordance with the present invention, an MO can be regarded as an arrangement of valence AO's which are determined by matching the color (or other indicia) characteristics of a "generator AO" at a central point within the molecule (or at the location of a central atom if the molecule has one) with the corresponding characteristics of the AO's at the non-central atom sites. In one embodiment, the opposite color (or other type of contrasting indicia) correspond to opposite signs of the wave functions of the electrons in a given AO or MO. The apparatus of this invention provides an "MO" and "AO" model system kit, which permits visualization by color, or other matching or mismatching relationships. By the novel central generator atomic orbital concept of this invention, it is relatively simple to generate MO's which duplicate the results arrived at by the far more complex group theory. The key generator AO's comprise a series of AO models used in sequence on a central probe. This novel principle, which underlies the method and apparatus of this invention, and its application in generating MO's are illustrated by the accompanying drawings and the following detailed specification.